fractions

FRACTION REVIEW
A.
INTRODUCTION
1.
What is a fraction?
Fraction = numerator
denominator
A fraction consists of a numerator (part) on top of a
denominator (total) separated by a horizontal line.
For example, the fraction of the circle which is shaded is:
2 (parts shaded)
4 (total parts)
In the square on the right, the fraction shaded is 3 and
8
5
the fraction unshaded is
8
2.
Equivalent Fractions – Multiplying
The three circles on the right each have
equal parts shaded, yet are represented
by different but equal fractions. These
fractions, because they are equal, are
called equivalent fractions.
4
8
2
4
1
2
Any fraction can be changed into an equivalent fraction
by multiplying both the numerator and denominator by the same number
1 x2
2 x2
=
2
4
Similarly
5 x 2 = 10
9 x2
18
or
1 x4 4
=
2 x4 8
so
124
2 4 8
or
5 x 3 = 15
9 x 3 27
so
5 = 10 = 15
9 18
27
You can see from the above examples that each fraction has an infinite number of fractions
that are equivalent to it.
Business Math Study Guide 2
Page 1
3.
Equivalent Fractions – Dividing (Reducing)
Equivalent fractions can also be created if both the
numerator and denominator can be divided by the
same number (a factor) evenly.
This process is called “reducing a fraction” by
dividing a common factor (a number which divides
into both the numerator and denominator evenly).
441
8 4 2
27  9  3
81 9 9
5 5 1
30 5 6
6
2 3
 
10 2 5
4.
Simplifying a Fraction (Reducing to its Lowest Terms)
It is usual to reduce a fraction until it can’t be reduced
any further.
A simplified fraction has no common factors which will
divide into both numerator and denominator.
Notice that, since 27 and 81 have a common factor of 9,
we find that 3 is an equivalent fraction.
9
But this fraction has a factor of 3 common to both
numerator and denominator.
So, we must reduce this fraction again. It is difficult to
see, but if we had known that 27 was a factor (divides
into both parts of the fraction evenly), we could have
arrived at the answer in one step
e.g.
8 8  1
24 8 3
27  9  3
81 9 9
331
9 3 3
27  27  1
81 27
3
45  15  3
60 15 4
Business Math Study Guide 2 – Fractions FB/2015
Page 2
5.
EXERCISE 1: Introduction to Fractions
a) Find the missing part of these equivalent fractions
1) 2 =
3
6
2)
3 =
4
12
3)
5 =
8
40
x2
Example:
3 =
5
10
x2
4)
1 =
32
16
5)
2 =
45
15
7)
7 =
10
100
8)
3 =
4
44
6.)
7 =
9
27
Since 5 x 2 = 10,
multiply the numerator
by 2, also.
So,
3 = 6
5
10
b) Find the missing part of these equivalent fractions.
÷5
1) 8 =
4
16
2)
24 =
9
27
Example:
5 =
2
10
÷5
3)
6 =
5
10
4)
5)
20 =
30
6
6) 90 =
100
50
25 =
35
7
Since 10 ÷ 5 = 2 divide
the numerator by 5, also.
So,
c) Simplify the following fractions (reduce to lowest terms).
1) 9
12
5) 20
25
9) 66
99
8
12
6) 14
21
10) 18
30
2)
Business Math Study Guide 2 – Fractions FB/2015
6
8
7) 8
16
3)
4) 15
20
8) 24
36
Page 3
5 = 1
2
10
B.
TYPES OF FRACTIONS
1. Common Fractions
A common fraction is one in which the numerator is less than the denominator
(or a fraction which is less than the number 1). A common fraction can also be called a proper fraction.
1,
2
e.g.
3,
4
88 , 8
93 15
are all common fractions.
2. Fractions that are Whole Numbers
Some fractions, when reduced, are really whole numbers (1, 2, 3, 4… etc).
Whole numbers occur if the denominator divides into the numerator evenly.
8 is the same as 8 ÷ 4 = 2 or 2
4
4
4
1
30 is the same as 30 ÷ 5 = 6 or 6
5
5
5
1
e.g.
So, the fraction 30 is really the whole number 6.
5
Notice that a whole number can always be written as a fraction with a denominator of 1.
e.g.
3.
10 = 10
1
Mixed Numbers
A mixed number is a combination of a whole number and a common fraction.
e.g.
23
5
(two and three-fifths)
27 2 (twenty-seven and two-ninths)
9
3
9
= 9 1 (always reduce fractions)
6
2
Business Math Study Guide 2 – Fractions FB/2015
Page 4
4. Improper Fractions
An improper fraction is one in which the numerator is larger than the
denominator.
From the circles on the right, we see that 1 3 (mixed number)
4
7
is the same as
(improper fraction).
4
An improper fraction, like 7 , can be changed to a mixed number by
4
dividing the denominator into the numerator and expressing the
remainder (3) as the numerator.
e.g.
16 = 3 1
5
5
29 = 3 5
8
8
13
4
1
= 4 7
7
4
4
3
14 = 4 2
3
3
7
4
=
= 13
4
A mixed number can be changed to an improper fraction by changing
the whole number to a fraction with the same denominator as the
common fraction.
2 3 = 10 and 3
5
5
5
= 13
5
10 1
9
=
=
90
9
91
9
and
A simple way to do this is to multiply the whole number by the denominator,
and then add the numerator.
e.g.
36  5
4x95
45 =
=
= 41
9
9
9
9
10
x
7

2
70

2
10 2 =
=
= 72
7
7
7
7
5. Simplifying fractions
All types of fractions must always be simplified (reduced to lowest terms).
e.g.
6 = 2,
9
3
2 5 = 21,
25
5
27 = 3 = 1 1
18
2
2
Note that many fractions can not be reduced since they have no common factors.
e.g.
17 , 4 , 18
21 9 37
Business Math Study Guide 2 – Fractions FB/2015
Page 5
1
9
6. EXERCISE 2 : Types of Fractions
a) Which of the following are common fractions (C), whole numbers (W), mixed numbers (M)
or improper fractions (I)?
1) 2
3
2) 3 4
5
3) 7
5
4) 8
8
5) 24
2
6) 5 8
19
7) 2 3
3
8) 25
24
9) 24
25
10) 12
12
4) 12
5
5) 100
99
6) 25
2
3) 8 2
3
4) 11 1
5
5) 9 4
5
6) 4 3
4
3) 2 12
18
4) 5 27
54
5) 25
15
6) 90
12
b) Change the following to mixed numbers:
1) 7
5
2) 18
11
3) 70
61
c) Change the following to improper fractions:
1) 2 1
5
2) 6 3
8
d) Simplify the following fractions:
1) 28
40
2) 80
10
Business Math Study Guide 2 – Fractions FB/2015
Page 6
C.
COMPARING FRACTIONS
In the diagram on the right, it is easy to see that 7 is larger
8
than 3 (since 7 is larger than 3).
8
However, it is not as easy to tell that 7 is larger than 5 .
8
6
In order to compare fractions, we must have the same (common)
7
8
=
3
8
denominators. This process is called
“Finding the Least Common Denominator” and is usually
abbreviated as finding the LCD or LCM (lowest common multiple).
Which is larger: 7 or 5 ?
8
6
In order to compare these fractions, we must change both fractions
to equivalent fractions with a common denominator.
To do this, take the largest denominator (8) and examine multiples
of it, until the other denominator (6) divides into it.
Notice that, when we multiply 8 x 3, we get 24, which 6 divides
into.
Now change the fractions to 24th s.
When we change these fractions to equivalent fractions with an
LCD of 24, we can easily see
that 7 is larger than 5 since 21 is greater than 20 .
24
24
8
6
Business Math Study Guide 2 – Fractions FB/2015
8x1 = 8
(6 doesn’t divide into 8)
8 x 2 = 16
(6 doesn’t divide into 16)
8 x 3 = 24 LCD
x3
x4
x3
x4
7 = 21
5 = 20
24
24
8
6
Page 7
Which is larger: 4 or 5 ?
9
12
Examine multiples of the larger denominator (12) until the smaller
denominator divides into it. This tells us that the LCD is 36.
Now, we change each fraction to equivalent fractions with the LCD
of 36.
x4
4 = 16
9
36
x4
12 x 1 = 12
12 x 2 = 24
12 x 3 = 36 (LCD)
LCD is 36.
x3
5 = 15
12
36
x3
So, 4 is larger than 5
9
12
_______________________________________________________________________________
13
Which is larger: 4 or
or 11 ?
5
12
15
Find the LCD by examining multiples of 15. Notice that, when we
multiply 15 x 4, we find that 60 is the number that all denominators
divide into.
x12
4 = 48
60
5
x12
x4
13 = 52
60
15
x4
15
15
15
15
x
x
x
x
1
2
3
4
=
=
=
=
x5
11 = 55
60
12
x5
So, 11 is the largest fraction.
12
Business Math Study Guide 2 – Fractions FB/2015
Page 8
15
30
45
60 (LCD)
Which is larger: 7 or 13 ?
9
18
Notice that one denominator (9) divides into the other denominator (18). This means that
the LCD = 18 and we only have to change one fraction  7  to an equivalent fraction.
9
x2
7 = 14
9
18
x2
So, 7 is larger than 13
9
18
__________________________________________________________
__________________________________________________________
1. EXERCISE 3: Comparing Fractions
Which is the largest fraction? (find LCD first)
1)
7 or 6
13
13
2) 1 9 or 18
10
10
3) 4 or 9
5
10
4)
3 or 5
13
12
5) 5 or 4
8
7
6) 1 or 6 or 7
11
2
12
8) 4 or 5 or 3
8
9
12
9) 1 or 3
4
16
7) 2 or 11
3
15
Business Math Study Guide 2 – Fractions FB/2015
Page 9
D.
ADDING FRACTIONS
There are four main operations that we can do with numbers: addition ( +
multiplication ( x ), and division ( ÷ ).
In order to add or subtract, fractions must have common denominators.
This is not required for multiplication or division.
), subtraction ( –
1. Adding with Common Denominators
To add fractions, if the denominators are the same, we simply add
the numerators and keep the same denominators.
e.g.
1
+ 2 = 3
4
4
4
Add 1 and 5
12
12
Since the denominators are common, simply add the numerators.
Notice that we must reduce the answer, if possible.
1 + 5 = 6
12 12
12
= 1
2
2. Adding When One Denominator is a Multiple of the Other
x3
Add
5
2
and
9
27
2 = 6
27
9
x3
Notice that the denominators are not common. Also notice that 27 is
a multiple of 9 (since 9 x 3 = 27). This means that the LCD = 27
(see the last example in “Comparing Fractions”).
2 + 5 = 6 + 5
27
27
27
9
11
=
27
Business Math Study Guide 2 – Fractions FB/2015
Page 10
),
3. Adding Any Fraction
Add 7 and 13
12
15
We must find a common denominator by examining multiples of the
largest denominator. We find that the LCD = 60.
7 + 13 = 35 + 52
60
60
12 15
= 87
60
= 1 9
20
3
Add 1 5 and 2
6
8
When adding mixed numbers, add the whole numbers and the
fractions separately. Find common denominators and add.
+
If an improper fraction occurs in the answer, change it to a common
fraction by doing the following.
1 5 = 1 20
24
6
3
2
= 2 9
8
24
total equals 3 29
24
3 29 = 3 + 1 5
24
24
= 4 5
24
4. The Language of Addition
1 + 2 CAN BE WORDED
2
3
1 plus 2
2
3
1 and 2
2
3
addition of 1 and 2
2
3
total of 1 and 2
2
3
1 combined with 2
2
3
sum of 1 and 2
2
3
1 more than (or greater than) 2
2
3
Note: All of these can be worded with the fractions in reverse order:
e.g. 2 plus 1 is the same as 1 plus 2
3
2
2
3
Business Math Study Guide 2 – Fractions FB/2015
Page 11
5.
EXERCISE 4: Adding Fractions
a) Add the following:
1)
1
2
+
5
5
2)
4
3
+
5
5
3)
4
2
+
9
9
4)
3
3
3
+
+
4
4
4
b) Find the sum of:
1)
2
1
+
3
9
2)
1
3
+
2
8
3)
1
5
+
4
16
4)
2
4
+
3
15
c) Add the following:
1
1
+ 4
2
4
2) 9
2
1
+ 3
3
6
1
4
+ 4
2
5
4) 2
3
1
+ 6
4
2
5) 4
1
5
+ 6
3
6
6) 6
1
3
+ 8
3
4
7) 7
2
4
+
3
5
8) 8
2
1
3
+ 6 + 1
3
4
8
1)
3
3) 8
d) Evaluate the following:
1)
2
3
and
3
7
2) total of
3)
1
1
plus
2
5
4)
5)
5
3
combined with
12
8
Business Math Study Guide 2 – Fractions FB/2015
5
3
and
6
8
3
5
greater than
2
7
6) sum of
1
3
and
6
14
Page 12
E.
SUBTRACTING FRACTIONS
1. Common Fractions
As in addition, we must have common denominators
in order to subtract. Find the LCD; change the fractions to
equivalent fraction with the LCD as the denominator. Then
subtract the numerators, but keep the same denominator.
5 3
8 8
=
2 3
3 8
=
2
1
or
8
4
16
9
24 24
7
=
24
2. Mixed Numbers
When subtracting whole numbers, subtract the whole numbers, and
then subtract the fractions separately.
3
5
3
2
-1 = 2
9
9
9
However, if the common fraction we are subtracting is smaller than
the other common fraction, we must borrow the number “1” from
the large whole number.
i.e.
4
2
7
2
9
=3+
+ , or 3
7
7
7
7
3
2
from 6 , first change the common fractions to
3
4
8
9
equivalent fractions with the LCD. Since
is smaller than
,
12
12
borrow from 6.
4
To subtract 1
6
8
12
8
20
= 5
+
= 5
12 12
12
12
Business Math Study Guide 2 – Fractions FB/2015
6
2
5
9
5
-2 = 3 -2
7
7
7
7
4
= 1
7
2
3
8
9
-1 = 6
-1
3
12
4
12
20
9
= 5
-1
12
12
11
= 4
12
Page 13
3. The Language of Subtraction
5 2
CAN BE WORDED
6 3
5
2
minus
6
3
(NOT
2
minus)
3
2 subtracted from 5
3
6
2
5
from
3
6
2
5
less than
3
6
5
2
decreased by or lowered by
6
3
the difference of
5
2
and
6
3
NOTE: Unlike addition, we can not reword the above with the fractions in reverse order:
1 2
2 1
i.e.
is NOT the same as
2 3
3 2
=============================================================================
4. EXERCISE 5: Subtracting Fractions
a) Subtract the following:
1)
9
1
12
8
2)
14
1
15
6
4)
7
2
9
3
5) 9
7)
3
5
4
8
8)
3)
2
1
- 6
3
6
5 3
6 8
6) 4
1
1
-1
2
4
3) 5
5
6
- 4
7
7
11
2
12
3
b) Subtract the following:
1)
6
4) 4
1
2
- 2
3
3
3
4
- 1
11
12
2) 13
5) 16
1
3
- 5
4
4
2
3
- 5
3
6) 9
4
1
3
- 4
6
8
c) Find the following:
1) What is
5
3
minus
?
8
16
2)
2
1
decreased by
is what?
7
21
3) What is
4
7
less than ?
9
9
4) What is
Business Math Study Guide 2 – Fractions FB/2015
1
9
from
?
6
24
Page 14
F.
MULTIPLYING FRACTIONS
1. Common Fractions
When multiplying fractions, a common denominator is not needed.
Simply multiply the numerators and multiply the denominators
separately.
Sometimes, we can reduce the fractions before multiplying.
2
5
25
10
x
=
=
3 9
3 9
27
1
3 5
3 5
3
x
=
=
5 7
7
5 7
1
Any common factor in either numerator can cancel with the same
factor in the denominator. Multiply after cancelling (reducing).
Note that any whole number (16) has the number “1” understood in
its denominator.
1
1
3
2
4 3
43
1
x
=
=
9 8
98
6
2
5
5  16
x 16 =
= 10
8
8 1
1
If more than two fractions are multiplied, the same principles apply.
2
3
1
2  3 1
1
x
x
=
=
9
20
4
9  20  4 120
2. Mixed Numbers
Mixed numbers must be changed to improper fractions before
3
multiplying. Remember that a mixed number (like 2 ) can be
4
changed to an improper fraction by multiplying the whole number
(2) by the denominator (4) and then adding the numerator.
See page 6 for instructions on changing a mixed number to an
improper fraction.
2
5
3
8
11 52
x 4
=
x
4
11
4
11
= 13
4
3
49 17
x2 =
x
9
7
9
7
119
9
2
or 13
9
=
Business Math Study Guide 2 – Fractions FB/2015
Page 15
3. The Language of Multiplication
1
2
x
CAN BE WORDED
2
3
1
2
multiplied by
2
3
1
2
by
2
3
1
2
of
2
3
the product of
1
2
and
2
3
NOTE: When multiplying, it doesn’t matter which fraction is first.
1
2
2
1
i.e.
x
is the same as
x
2
3
3
2
=============================================================================
4. EXERCISE 6: Multiplying Fractions
a) Multiply (cancel first, when possible):
1)
2
3
x
3
4
2)
12
8
x
9
16
3)
5
9
x
3 15
4)
2
15
x
5
21
5)
5
48
x
8
125
6)
3 16
9
x
x
4
27 16
7)
5
7
16
x
x 90 x 20 x
3
4
14
8) 2
10)
18 57
6
8 2
x
x
x x
19
4
32 9 3
4
4
x 3
5
7
9) 8
12
1
x2
25
37
12) 2
11) 1
2 15
x
3 21
5
x 6
12
b) Find the following:
1) What is
4)
4
5
of 2
1
7
2
of 45?
3
is what number?
2)
41
5
27
by 14 is what number? 3) What is times
?
9
10
7
5)
1
62
of
is what number?
2
63
Business Math Study Guide 2 – Fractions FB/2015
Page 16
G.
DIVIDING FRACTIONS
To divide fractions, we invert (take the reciprocal of) the fraction that
we are dividing by, then cancel (reduce), and then multiply.
Taking the reciprocal of a fraction involves “flipping” the fraction so
that the numerator and denominator switch places.
Note that a whole number is really a fraction (e.g. 4 =
4
).
1
2
3
reciprocal
3
2
8
19
reciprocal
19
8
4 reciprocal
4
1
1. Common Fractions
Simply invert (take the reciprocal of) the fractions that we are
8
dividing by ( ). Then cancel and multiply.
9
Note: you can only cancel after the division is changed to a
multiplication.
5 8
5 9
45
 =  =
7 9
7 8 56
9
3
9 32
=
=6


16 32 16 3
2. Mixed Numbers
As in multiplication, mixed numbers must be changed to improper
fractions.
Business Math Study Guide 2 – Fractions FB/2015
2
1
5 15
1 2 = 
3
7
3 7
5 7
= 
3 15
7
=
9
Page 17
3. The Language of Division
1
2
CAN BE WORDED

2
3
1
2
divided by
2
3
2
3
into
1
2
divide
1
2
by
2
3
NOTE: In multiplication, the order of the fractions was not important.
1
2
2
1
i.e.
x
is the same as
x
2
3
3
2
In division, this is not the case. The order of the fractions is important.
Consider the following:
1 2
1 3
3
 =  =
2 3
2 2
4
but
2 1
2 2
4
1
=1
 =  =
3 2
3 1
3
3
=============================================================================
4. EXERCISE 7: Dividing Fractions
a) Divide:
1)
3 9

5 15
2)
3 12

7 5
3) 6 
4)
7
 14
12
5)
13 39

15 40
6) 3
7)
27
1
8
30
3
7
2
8) 1  1
9
3
10) 2
2
3
3 11

10 15
2
1
9) 4  3
3
2
3
9
3
11
22
b) Find the following:
1)
4
2
divided by
5
5
2)
2
4
divided by
5
5
Business Math Study Guide 2 – Fractions FB/2015
3)
9
12
into
4
25
Page 18
4) Divide 1
2
2
by
3
3
5)
4
4
into 1
9
5
Business Math Study Guide 2 – Fractions FB/2015
6) Divide
1
2
by
4
5
Page 19
FRACTION REVIEW:
Decide which operation ( +, -, x,  ) by the wording in the question. Then find the answer.
3
of 40?
5
1
2
3) How much is
from ?
2
3
4
2
5)
and
equals….?
5
3
9
7) What is
of 300?
10
3
8
9)
from equals….?
4
9
5
21
11) How much is
of
?
7
50
3
2
13)
of
equals….?
4
3
1) What is
2) How much is 4
4) How much is
2
1
from 6 ?
3
5
3
of 21?
7
2
divided by 14 is what number?
3
2
8) What is
into 12?
5
2
3
10) What is 1 by 3
3
8?
2
1
4
12) Find the total of
and
and ?
3
6
9
2
3
14) What is
greater than 4 ?
5
5
6)
Business Math Study Guide 2 – Fractions FB/2015
Page 20
ANSWER KEY – FRACTION REVIEW
EXERCISE 1: Introduction to Fractions (Page 3)
a)
1)
4
6
2)
9
12
3)
25
40
4)
2
32
5)
6
45
6)
21
27
b)
1)
2
4
2)
8
9
3)
3
5
4)
5
7
5)
4
6
6)
45
50
c)
1)
3
4
2)
2
3
3)
3
4
4)
3
4
5)
4
5
6)
2
3
7)
70
100
8)
33
44
7)
1
2
8)
2
3
9)
2
3
10)
3
5
EXERCISE 2: Types of Fractions (Page 6)
a)
1) C
2) M
b)
1) 1
c)
1)
11
5
2)
d)
1)
7
10
2) 8
2
5
2) 1
3) I
4) W
7
9
2
3) 1
4) 2
61
5
11
51
8
3)
26
3
3) 2
2
3
4)
5) W
5) 1
6) M
8) I
9) C
10) W
1
1
6) 12
99
2
56
5
5)
49
5
1
2
5)
5
2
or 1
3
3
4) 5
7) W
6)
19
4
6)
15
1
or 7
2
2
6)
7
12
EXERCISE 3: Comparing Fractions (Page 9)
1)
7
13
2) 1
9
10
3)
9
10
4)
5
12
5)
5
8
11
15
7)
8)
4
9
9)
1
4
EXERCISE 4: Adding Fractions (Page 12)
a)
1)
3
5
2)
7
2
2
or 1
3)
5
3
5
4)
9
1
or 2
4
4
b)
1)
7
9
2)
7
8
4)
14
15
c)
1) 7
3
4
2) 12
d)
1) 1
2
21
2) 1
3)
5
6
5
24
9
16
3) 13
3)
7
10
3
10
4) 9
1
4
5) 11
4) 2
3
14
5)
Business Math Study Guide 2 – Fractions FB/2015
1
6
19
24
6) 15
6)
1
12
7) 8
8
21
Page 21
7
15
8) 16
7
24
EXERCISE 5: Subtracting Fractions (Page 14)
5
8
a)
1)
b)
1) 3
c)
1)
2)
2
3
7
16
23
30
3)
11
24
4)
1
2
3)
6
7
4) 2
5
21
3)
1
3
4)
2) 7
2)
1
9
5) 3
5
6
1
2
5) 10
11
12
6) 3
1
4
6) 4
19
24
7)
1
8
5
24
EXERCISE 6: Multiplying Fractions (Page 16)
a)
b)
1
2
2)
2
3
8) 10
9)
130
4
or 6
21
21
1) 30
2) 82
1)
3) 1
3)
4)
2
7
5)
10) 1
3
1
or 1
2
2
4)
1
2
6
25
11) 3
6)
1
4
12)
12
5
or 1
7
7
5)
7) 6000
29
1
or 14
2
2
31
63
EXERCISE 7: Dividing Fractions (Page 18)
a)
1) 1
8) 1
b)
1
15
1) 2
2)
5
28
3) 9
1
3
10)
9) 1
2)
1
2
3)
4)
1
24
5)
8
9
6) 4
1
2
7)
27
250
2
3
16
75
4) 2
1
2
1
20
6)
5
8
7
15
6)
1
21
7) 270
1
2
14) 5
5) 4
Fraction Review (Page 19)
a)
1) 24
2) 1
8) 30
9)
8
15
5
36
3)
1
6
10) 5
4) 9
5
8
11)
5) 1
3
10
Business Math Study Guide 2 – Fractions FB/2015
12) 1
5
18
13)
Page 22
8)
1
4